3.3.5 \(\int \frac {-33750 x^2-101250 x^3+379586250 x^4-33750 x^5+(6+30 x-67440 x^2-134940 x^3-67470 x^4+6 x^5) \log (x)}{x+5 x^2+10 x^3+10 x^4+5 x^5+x^6} \, dx\) [205]

Optimal. Leaf size=19 \[ 3 \left (\frac {5625 x^2}{(1+x)^2}-\log (x)\right )^2 \]

[Out]

3*(225*x^2/(1/5*x+1/5)^2-ln(x))^2

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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(58\) vs. \(2(19)=38\).
time = 0.29, antiderivative size = 58, normalized size of antiderivative = 3.05, number of steps used = 14, number of rules used = 10, integrand size = 77, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {6820, 12, 6874, 1634, 2404, 2338, 2356, 46, 2351, 31} \begin {gather*} -\frac {379687500}{x+1}+\frac {569531250}{(x+1)^2}-\frac {379687500}{(x+1)^3}+\frac {94921875}{(x+1)^4}+3 \log ^2(x)-\frac {67500 x \log (x)}{x+1}-\frac {33750 \log (x)}{(x+1)^2}+33750 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-33750*x^2 - 101250*x^3 + 379586250*x^4 - 33750*x^5 + (6 + 30*x - 67440*x^2 - 134940*x^3 - 67470*x^4 + 6*
x^5)*Log[x])/(x + 5*x^2 + 10*x^3 + 10*x^4 + 5*x^5 + x^6),x]

[Out]

94921875/(1 + x)^4 - 379687500/(1 + x)^3 + 569531250/(1 + x)^2 - 379687500/(1 + x) + 33750*Log[x] - (33750*Log
[x])/(1 + x)^2 - (67500*x*Log[x])/(1 + x) + 3*Log[x]^2

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 46

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x
)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && Lt
Q[m + n + 2, 0])

Rule 1634

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)
^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2]) &&
GtQ[Expon[Px, x], 2]

Rule 2338

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rule 2351

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^(r_.))^(q_), x_Symbol] :> Simp[x*(d + e*x^r)^(q +
1)*((a + b*Log[c*x^n])/d), x] - Dist[b*(n/d), Int[(d + e*x^r)^(q + 1), x], x] /; FreeQ[{a, b, c, d, e, n, q,
r}, x] && EqQ[r*(q + 1) + 1, 0]

Rule 2356

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[(d + e*x)^(q + 1)
*((a + b*Log[c*x^n])^p/(e*(q + 1))), x] - Dist[b*n*(p/(e*(q + 1))), Int[((d + e*x)^(q + 1)*(a + b*Log[c*x^n])^
(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && GtQ[p, 0] && NeQ[q, -1] && (EqQ[p, 1] || (Integers
Q[2*p, 2*q] &&  !IGtQ[q, 0]) || (EqQ[p, 2] && NeQ[q, 1]))

Rule 2404

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*x^
n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, n}, x] && RationalFunctionQ[RFx, x] && IGtQ[p, 0]

Rule 6820

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rule 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {6 \left (1+3 x-11247 x^2+x^3\right ) \left (-5625 x^2+(1+x)^2 \log (x)\right )}{x (1+x)^5} \, dx\\ &=6 \int \frac {\left (1+3 x-11247 x^2+x^3\right ) \left (-5625 x^2+(1+x)^2 \log (x)\right )}{x (1+x)^5} \, dx\\ &=6 \int \left (-\frac {5625 x \left (1+3 x-11247 x^2+x^3\right )}{(1+x)^5}+\frac {\left (1+3 x-11247 x^2+x^3\right ) \log (x)}{x (1+x)^3}\right ) \, dx\\ &=6 \int \frac {\left (1+3 x-11247 x^2+x^3\right ) \log (x)}{x (1+x)^3} \, dx-33750 \int \frac {x \left (1+3 x-11247 x^2+x^3\right )}{(1+x)^5} \, dx\\ &=6 \int \left (\frac {\log (x)}{x}+\frac {11250 \log (x)}{(1+x)^3}-\frac {11250 \log (x)}{(1+x)^2}\right ) \, dx-33750 \int \left (\frac {11250}{(1+x)^5}-\frac {33750}{(1+x)^4}+\frac {33750}{(1+x)^3}-\frac {11251}{(1+x)^2}+\frac {1}{1+x}\right ) \, dx\\ &=\frac {94921875}{(1+x)^4}-\frac {379687500}{(1+x)^3}+\frac {569531250}{(1+x)^2}-\frac {379721250}{1+x}-33750 \log (1+x)+6 \int \frac {\log (x)}{x} \, dx+67500 \int \frac {\log (x)}{(1+x)^3} \, dx-67500 \int \frac {\log (x)}{(1+x)^2} \, dx\\ &=\frac {94921875}{(1+x)^4}-\frac {379687500}{(1+x)^3}+\frac {569531250}{(1+x)^2}-\frac {379721250}{1+x}-\frac {33750 \log (x)}{(1+x)^2}-\frac {67500 x \log (x)}{1+x}+3 \log ^2(x)-33750 \log (1+x)+33750 \int \frac {1}{x (1+x)^2} \, dx+67500 \int \frac {1}{1+x} \, dx\\ &=\frac {94921875}{(1+x)^4}-\frac {379687500}{(1+x)^3}+\frac {569531250}{(1+x)^2}-\frac {379721250}{1+x}-\frac {33750 \log (x)}{(1+x)^2}-\frac {67500 x \log (x)}{1+x}+3 \log ^2(x)+33750 \log (1+x)+33750 \int \left (\frac {1}{-1-x}+\frac {1}{x}-\frac {1}{(1+x)^2}\right ) \, dx\\ &=\frac {94921875}{(1+x)^4}-\frac {379687500}{(1+x)^3}+\frac {569531250}{(1+x)^2}-\frac {379687500}{1+x}+33750 \log (x)-\frac {33750 \log (x)}{(1+x)^2}-\frac {67500 x \log (x)}{1+x}+3 \log ^2(x)\\ \end {aligned} \end {gather*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(41\) vs. \(2(19)=38\).
time = 0.05, size = 41, normalized size = 2.16 \begin {gather*} 3 \left (-\frac {31640625 \left (1+4 x+6 x^2+4 x^3\right )}{(1+x)^4}-\frac {11250 x^2 \log (x)}{(1+x)^2}+\log ^2(x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-33750*x^2 - 101250*x^3 + 379586250*x^4 - 33750*x^5 + (6 + 30*x - 67440*x^2 - 134940*x^3 - 67470*x^
4 + 6*x^5)*Log[x])/(x + 5*x^2 + 10*x^3 + 10*x^4 + 5*x^5 + x^6),x]

[Out]

3*((-31640625*(1 + 4*x + 6*x^2 + 4*x^3))/(1 + x)^4 - (11250*x^2*Log[x])/(1 + x)^2 + Log[x]^2)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(58\) vs. \(2(21)=42\).
time = 0.07, size = 59, normalized size = 3.11

method	result	size
default	\(\frac {569531250}{\left (x +1\right )^{2}}+\frac {94921875}{\left (x +1\right )^{4}}-\frac {379687500}{x +1}-\frac {379687500}{\left (x +1\right )^{3}}+\frac {33750 \ln \left (x \right ) x \left (2+x \right )}{\left (x +1\right )^{2}}+3 \ln \left (x \right )^{2}-\frac {67500 \ln \left (x \right ) x}{x +1}\)	\(59\)
norman	\(\frac {5625 \ln \left (x \right )[phone] x^{4}-45000 x^{3} \ln \left (x \right )-28125 x^{4} \ln \left (x \right )+22500 x \ln \left (x \right )+3 \ln \left (x \right )^{2}+12 x \ln \left (x \right )^{2}+18 x^{2} \ln \left (x \right )^{2}+12 x^{3} \ln \left (x \right )^{2}+3 x^{4} \ln \left (x \right )^{2}}{\left (x +1\right )^{4}}-5625 \ln \left (x \right )\)	\(81\)
risch	\(3 \ln \left (x \right )^{2}+\frac {33750 \left (2 x +1\right ) \ln \left (x \right )}{x^{2}+2 x +1}-\frac {16875 \left (2 x^{4} \ln \left (x \right )+8 x^{3} \ln \left (x \right )+12 x^{2} \ln \left (x \right )+22500 x^{3}+8 x \ln \left (x \right )+33750 x^{2}+2 \ln \left (x \right )+22500 x +5625\right )}{\left (x^{2}+2 x +1\right )^{2}}\)	\(84\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((6*x^5-67470*x^4-134940*x^3-67440*x^2+30*x+6)*ln(x)-33750*x^5+379586250*x^4-101250*x^3-33750*x^2)/(x^6+5*
x^5+10*x^4+10*x^3+5*x^2+x),x,method=_RETURNVERBOSE)

[Out]

569531250/(x+1)^2+94921875/(x+1)^4-379687500/(x+1)-379687500/(x+1)^3+33750*ln(x)*x*(2+x)/(x+1)^2+3*ln(x)^2-675
00*ln(x)*x/(x+1)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 172 vs. \(2 (19) = 38\).
time = 0.34, size = 172, normalized size = 9.05 \begin {gather*} -\frac {5625 \, {\left (48 \, x^{3} + 108 \, x^{2} + 88 \, x + 25\right )}}{2 \, {\left (x^{4} + 4 \, x^{3} + 6 \, x^{2} + 4 \, x + 1\right )}} - \frac {189793125 \, {\left (4 \, x^{3} + 6 \, x^{2} + 4 \, x + 1\right )}}{2 \, {\left (x^{4} + 4 \, x^{3} + 6 \, x^{2} + 4 \, x + 1\right )}} + \frac {16875 \, {\left (6 \, x^{2} + 4 \, x + 1\right )}}{2 \, {\left (x^{4} + 4 \, x^{3} + 6 \, x^{2} + 4 \, x + 1\right )}} - \frac {3 \, {\left (11250 \, x^{2} \log \left (x\right ) - {\left (x^{2} + 2 \, x + 1\right )} \log \left (x\right )^{2} - 11250 \, x - 11250\right )}}{x^{2} + 2 \, x + 1} + \frac {5625 \, {\left (4 \, x + 1\right )}}{2 \, {\left (x^{4} + 4 \, x^{3} + 6 \, x^{2} + 4 \, x + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6*x^5-67470*x^4-134940*x^3-67440*x^2+30*x+6)*log(x)-33750*x^5+379586250*x^4-101250*x^3-33750*x^2)/
(x^6+5*x^5+10*x^4+10*x^3+5*x^2+x),x, algorithm="maxima")

[Out]

-5625/2*(48*x^3 + 108*x^2 + 88*x + 25)/(x^4 + 4*x^3 + 6*x^2 + 4*x + 1) - 189793125/2*(4*x^3 + 6*x^2 + 4*x + 1)
/(x^4 + 4*x^3 + 6*x^2 + 4*x + 1) + 16875/2*(6*x^2 + 4*x + 1)/(x^4 + 4*x^3 + 6*x^2 + 4*x + 1) - 3*(11250*x^2*lo
g(x) - (x^2 + 2*x + 1)*log(x)^2 - 11250*x - 11250)/(x^2 + 2*x + 1) + 5625/2*(4*x + 1)/(x^4 + 4*x^3 + 6*x^2 + 4
*x + 1)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 77 vs. \(2 (19) = 38\).
time = 0.39, size = 77, normalized size = 4.05 \begin {gather*} -\frac {3 \, {\left (126562500 \, x^{3} - {\left (x^{4} + 4 \, x^{3} + 6 \, x^{2} + 4 \, x + 1\right )} \log \left (x\right )^{2} + 189843750 \, x^{2} + 11250 \, {\left (x^{4} + 2 \, x^{3} + x^{2}\right )} \log \left (x\right ) + 126562500 \, x + 31640625\right )}}{x^{4} + 4 \, x^{3} + 6 \, x^{2} + 4 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6*x^5-67470*x^4-134940*x^3-67440*x^2+30*x+6)*log(x)-33750*x^5+379586250*x^4-101250*x^3-33750*x^2)/
(x^6+5*x^5+10*x^4+10*x^3+5*x^2+x),x, algorithm="fricas")

[Out]

-3*(126562500*x^3 - (x^4 + 4*x^3 + 6*x^2 + 4*x + 1)*log(x)^2 + 189843750*x^2 + 11250*(x^4 + 2*x^3 + x^2)*log(x
) + 126562500*x + 31640625)/(x^4 + 4*x^3 + 6*x^2 + 4*x + 1)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (19) = 38\).
time = 0.12, size = 61, normalized size = 3.21 \begin {gather*} \frac {\left (67500 x + 33750\right ) \log {\left (x \right )}}{x^{2} + 2 x + 1} - \frac {379687500 x^{3} + 569531250 x^{2} + 379687500 x + 94921875}{x^{4} + 4 x^{3} + 6 x^{2} + 4 x + 1} + 3 \log {\left (x \right )}^{2} - 33750 \log {\left (x \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6*x**5-67470*x**4-134940*x**3-67440*x**2+30*x+6)*ln(x)-33750*x**5+379586250*x**4-101250*x**3-33750
*x**2)/(x**6+5*x**5+10*x**4+10*x**3+5*x**2+x),x)

[Out]

(67500*x + 33750)*log(x)/(x**2 + 2*x + 1) - (379687500*x**3 + 569531250*x**2 + 379687500*x + 94921875)/(x**4 +
4*x**3 + 6*x**2 + 4*x + 1) + 3*log(x)**2 - 33750*log(x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6*x^5-67470*x^4-134940*x^3-67440*x^2+30*x+6)*log(x)-33750*x^5+379586250*x^4-101250*x^3-33750*x^2)/
(x^6+5*x^5+10*x^4+10*x^3+5*x^2+x),x, algorithm="giac")

[Out]

integrate(-6*(5625*x^5 - 63264375*x^4 + 16875*x^3 + 5625*x^2 - (x^5 - 11245*x^4 - 22490*x^3 - 11240*x^2 + 5*x
+ 1)*log(x))/(x^6 + 5*x^5 + 10*x^4 + 10*x^3 + 5*x^2 + x), x)

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Mupad [B]
time = 0.52, size = 28, normalized size = 1.47 \begin {gather*} \frac {3\,{\left (\ln \left (x\right )+x^2\,\ln \left (x\right )+2\,x\,\ln \left (x\right )-5625\,x^2\right )}^2}{{\left (x+1\right )}^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(33750*x^2 - log(x)*(30*x - 67440*x^2 - 134940*x^3 - 67470*x^4 + 6*x^5 + 6) + 101250*x^3 - 379586250*x^4
+ 33750*x^5)/(x + 5*x^2 + 10*x^3 + 10*x^4 + 5*x^5 + x^6),x)

[Out]

(3*(log(x) + x^2*log(x) + 2*x*log(x) - 5625*x^2)^2)/(x + 1)^4

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